Connection (principal bundle)

In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.

A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.

Formal definition

Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

Connection (affine bundle)

Let be an affine bundle modelled over a vector bundle . A connection on is called the affine connection if it as a section of the jet bundle of is an affine bundle morphism over . In particular, this is the case of an affine connection on the tangent bundle of a smooth manifold .

With respect to affine bundle coordinates on , an affine connection on is given by the tangent-valued connection form

An affine bundle is a fiber bundle with a general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection , the corresponding linear derivative of an affine morphism defines a unique linear connection on a vector bundle . With respect to linear bundle coordinates on , this connection reads

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If is a vector bundle, both an affine connection and an associated linear connection are connections on the same vector bundle , and their difference is a basic soldering form on . Thus, every affine connection on a vector bundle is a sum of a linear connection and a basic soldering form on .

Connection (Elastica song)

"Connection" is a song released by the Britpop group Elastica. It was originally released in 1994 as a single and the album version was not released until 1995 on their self-titled debut.

The song was the subject of controversy, due to its overt similarity to another band's work. The intro synthesizer part (later repeated as a guitar figure) is lifted from the guitar riff in Wire's "Three Girl Rhumba" and transposed down a semitone. A judgment resulted in an out-of-court settlement and the credits were rewritten.

The song is the theme to the UK television programme Trigger Happy TV. A live version of the song was featured on the MuchMusic live compilation album, Much at Edgefest '99.

The song was covered by Elastica's label-mates Collapsed Lung and released as a seven-inch vinyl single in 1995.

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Microphone

A microphone, colloquially mic or mike (/ˈmaɪk/), is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. Electromagnetic transducers facilitate the conversion of acoustic signals into electrical signals. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and public events, motion picture production, live and recorded audio engineering, two-way radios, megaphones, radio and television broadcasting, and in computers for recording voice, speech recognition, VoIP, and for non-acoustic purposes such as ultrasonic checking or knock sensors.

Most microphones today use electromagnetic induction (dynamic microphones), capacitance change (condenser microphones) or piezoelectricity (piezoelectric microphones) to produce an electrical signal from air pressure variations. Microphones typically need to be connected to a preamplifier before the signal can be amplified with an audio power amplifier and a speaker or recorded.

2.0 (98 Degrees album)

2.0 is the fourth studio album by American pop group 98 Degrees. The album was released on May 7, 2013 through eOne Music. It is their first studio album in thirteen years following Revelation (2000), and their first released through an independent record label.2.0 debuted at number 65 on the Billboard 200, becoming the group's lowest-charting album since their 1997 debut album, 98 Degrees which debuted at number 145.

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Microphone (Darin song)

Microphone is a song by Darin Zanyar from his album Lovekiller. The song was not released as an official single but due to strong digital downloads of the song upon the release of Darin's album Lovekiller on August 18, 2010, the song entered on Swedish Charts at number 16 on the week of August 27, 2010. The song peaked at number 15 on the third week.

Microphone was written and produced by Darin and Tony Nilsson. Microphone was first heard live on August 14, 2010 when Darin performed the song on Sommarkrysset 2010. Darin performed the song across Sweden to promote the album Lovekiller. The song had been used in a promotional video for the 2010 season of Idol, an equivalent of American Idol in the United States.

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Military reserve

A military reserve, reserve formation, or simply reserve, is a group of military personnel or units which are initially not committed to a battle by their commander so that they are available to address unforeseen situations or exploit suddenly developing opportunities. Such forces may be held back to defend against attack from other enemy forces, to be committed to the existing battle if the enemy exposes a vulnerability, or to serve as relief for troops already fighting. Some of the different categories of military reserves are: tactical reserve, operational reserve, strategic reserve.

A military reserve is different from a military reserve force, which is a military organization composed of military personnel who maintain their military skills and readiness in a long-term part-time commitment to support their country if needed. Military reserve refers to specific trained pre-organized forces operating as an on call basis from the main military force.

Background

The concept of reserve forces dates at least to early Roman legionary formations. In the early Roman legion, the soldiers were divided by wealth and age, with the poorer and younger Hastati forming the first line, followed by the Principes, and the wealthier and older Triarii. The first two lines were expected to do the majority of the fighting, with the third line held in reserve, both as a last resort and to deter routing.